The invention relates generally to the art of computer graphics and the modeling of objects. More particularly the invention relates to the modeling of objects using subdivision surfaces and to the modeling of objects with semi-sharp creases or edges.
No real objects have perfectly sharp features. All edges, even those on seemingly sharp objects, have some finite radius of curvature. Though inevitable in real life, this unavoidable lack of precision presents a difficult problem for computer graphics and computer animation. Some modeling methods, e.g., the use of piecewise linear surfaces (polygon meshes) work well for objects with sharp boundaries. Other methods, e.g., NURBS (non-uniform rational B-splines) work well (i.e., are more accurate and compact) for modeling curved surfaces, but fair less well and are less efficient for modeling objects with sharp features.
In recent work, Hoppe, et al. have shown that piecewise smooth surfaces incorporating sharp features, including edges, creases, darts and corners, can be efficiently modeled using subdivision surfaces by altering the standard Loop subdivision rules in the region of such sharp features. Hoppe, et al., Piecewise Smooth Surface Reconstruction, Computer Graphics (SIGGRAPH ""94 Proceedings), pgs. 295-302. The modified subdivision surface technique developed by Hoppe, et al, provides for an efficient method for modeling objects containing both curved surfaces and sharp features. The resulting sharp features are, however, infinitely sharp, i.e., the tangent plane is discontinuous across the sharp feature.
Even the method of Hoppe, et al. does not, therefore, solve the problem of modeling real objects with finite radius edges, creases and corners. To model such objects with existing techniques, either subdivision surfaces, NURBS, or polygons, requires vastly complicating the model by including many closely spaced control points or polygons in the region of the finite radius contour. As soon as one moves away from infinite sharpness, which can be modeled easily and efficiently with subdivision surfaces following Hoppe, et al. most, if not all of the advantages of the method are lost, and one must create a vastly more complicated model to enjoy the incremental enhancement in realism. Accordingly, there is a need for a way to efficiently model objects with semi-sharp features and, more generally, there is a need for a way to sculpt the limit surface defined by subdivision without complicating the initial mesh.
The present invention involves a method for modeling objects with surfaces created by sequentially combining different subdivision rules. By subdividing a mesh a finite number of times with one or more sets of xe2x80x9cspecial rulesxe2x80x9d before taking the infinite subdivision limit with xe2x80x9cstandard rulesxe2x80x9d one can produce different limit surfaces from the same initial mesh.
One important application of the invention is to the modeling of objects with smooth and semi-sharp features. The present invention allows one to model objects with edges and creases of continuously variable sharpness without adding vertices or otherwise changing the underlying control point mesh. The result is achieved in the exemplary embodiment by explicitly subdividing the initial mesh using the Hoppe, et al. rules or a variation which like the Hoppe, et al. rules do not require tangent plain continuity in the region of a sharp feature. After a finite number of iterations, one switches to the traditional continuous tangent plane rules for successive iterations. On may then push the final mesh points to their smooth surface infinite subdivision limits. The number of iterations performed with the xe2x80x9csharpxe2x80x9d (i.e., discontinuous tangent plane) rules determines the sharpness of the feature on the limit surface. Though the number of explicit subdivisions must be an integer, xe2x80x9cfractional smoothnessxe2x80x9d can be achieved by interpolating the position of points between the locations determined using N and N+1 iterations with the sharp rules.
In another exemplary embodiment, the invention is used to improve the shape of Catmull-Clark limit surfaces derived from initial meshes that include triangular faces.